If the simple graph G has 20 vertices and 55 edges, how many edges does its complement G' has?

Sure, here is the step by step solution:

1. First, we need to know the maximum number of edges in a simple graph with 20 vertices. This can be calculated using the formula n(n-1)/2, where n is the number of vertices. So, the maximum number of edges is 20(20-1)/2 = 190.

2. The complement of a graph G is a graph G' on the same vertices such that two distinct vertices of G' are adjacent if and only if they are not adjacent in G. This means that G' will have all the edges that G does not have.

3. So, to find the number of edges in G', we subtract the number of edges in G from the maximum number of edges. That is 190 - 55 = 135.

So, the complement of the graph G has 135 edges.